Average Error: 1.7 → 0.7
Time: 1.4min
Precision: binary64
\[z \leq 0.5\]
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
↓
\[\begin{array}{l} t_0 := \frac{676.5203681218851}{1 - z}\\ t_1 := \left(1 - z\right) - 1\\ t_2 := t_1 + 7\\ t_3 := \mathsf{fma}\left(t_0, t_0 + -0.9999999999998099, 0.9999999999996197\right)\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(t_2 + 0.5\right)}^{\left(0.5 + \log \left(\frac{e^{1 - z}}{e}\right)\right)}\right) \cdot e^{-0.5 - t_2}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2 - z, 0.9999999999994297 + {t_0}^{3}, -1259.1392167224028 \cdot t_3\right)\right)\right)}{\left(2 - z\right) \cdot t_3} + \frac{771.3234287776531}{t_1 + 3}\right) + \frac{-176.6150291621406}{t_1 + 4}\right) + \frac{12.507343278686905}{t_1 + 5}\right) + \frac{-0.13857109526572012}{t_1 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_2}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t_1 + 8}\right)\right) \end{array} \]
\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)
↓
\begin{array}{l}
t_0 := \frac{676.5203681218851}{1 - z}\\
t_1 := \left(1 - z\right) - 1\\
t_2 := t_1 + 7\\
t_3 := \mathsf{fma}\left(t_0, t_0 + -0.9999999999998099, 0.9999999999996197\right)\\
\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(t_2 + 0.5\right)}^{\left(0.5 + \log \left(\frac{e^{1 - z}}{e}\right)\right)}\right) \cdot e^{-0.5 - t_2}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2 - z, 0.9999999999994297 + {t_0}^{3}, -1259.1392167224028 \cdot t_3\right)\right)\right)}{\left(2 - z\right) \cdot t_3} + \frac{771.3234287776531}{t_1 + 3}\right) + \frac{-176.6150291621406}{t_1 + 4}\right) + \frac{12.507343278686905}{t_1 + 5}\right) + \frac{-0.13857109526572012}{t_1 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_2}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t_1 + 8}\right)\right)
\end{array}

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (*
    (*
     (sqrt (* PI 2.0))
     (pow (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5) (+ (- (- 1.0 z) 1.0) 0.5)))
    (exp (- (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5))))
   (+
    (+
     (+
      (+
       (+
        (+
         (+
          (+
           0.9999999999998099
           (/ 676.5203681218851 (+ (- (- 1.0 z) 1.0) 1.0)))
          (/ -1259.1392167224028 (+ (- (- 1.0 z) 1.0) 2.0)))
         (/ 771.3234287776531 (+ (- (- 1.0 z) 1.0) 3.0)))
        (/ -176.6150291621406 (+ (- (- 1.0 z) 1.0) 4.0)))
       (/ 12.507343278686905 (+ (- (- 1.0 z) 1.0) 5.0)))
      (/ -0.13857109526572012 (+ (- (- 1.0 z) 1.0) 6.0)))
     (/ 9.984369578019572e-6 (+ (- (- 1.0 z) 1.0) 7.0)))
    (/ 1.5056327351493116e-7 (+ (- (- 1.0 z) 1.0) 8.0))))))
↓
(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ 676.5203681218851 (- 1.0 z)))
        (t_1 (- (- 1.0 z) 1.0))
        (t_2 (+ t_1 7.0))
        (t_3 (fma t_0 (+ t_0 -0.9999999999998099) 0.9999999999996197)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (*
      (*
       (sqrt (* PI 2.0))
       (pow (+ t_2 0.5) (+ 0.5 (log (/ (exp (- 1.0 z)) E)))))
      (exp (- -0.5 t_2)))
     (+
      (+
       (+
        (+
         (+
          (+
           (/
            (expm1
             (log1p
              (fma
               (- 2.0 z)
               (+ 0.9999999999994297 (pow t_0 3.0))
               (* -1259.1392167224028 t_3))))
            (* (- 2.0 z) t_3))
           (/ 771.3234287776531 (+ t_1 3.0)))
          (/ -176.6150291621406 (+ t_1 4.0)))
         (/ 12.507343278686905 (+ t_1 5.0)))
        (/ -0.13857109526572012 (+ t_1 6.0)))
       (/ 9.984369578019572e-6 t_2))
      (/ 1.5056327351493116e-7 (+ t_1 8.0)))))))
double code(double z) {
	return (((double) M_PI) / sin(((double) M_PI) * z)) * (((sqrt(((double) M_PI) * 2.0) * pow(((((1.0 - z) - 1.0) + 7.0) + 0.5), (((1.0 - z) - 1.0) + 0.5))) * exp(-((((1.0 - z) - 1.0) + 7.0) + 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (((1.0 - z) - 1.0) + 1.0))) + (-1259.1392167224028 / (((1.0 - z) - 1.0) + 2.0))) + (771.3234287776531 / (((1.0 - z) - 1.0) + 3.0))) + (-176.6150291621406 / (((1.0 - z) - 1.0) + 4.0))) + (12.507343278686905 / (((1.0 - z) - 1.0) + 5.0))) + (-0.13857109526572012 / (((1.0 - z) - 1.0) + 6.0))) + (9.984369578019572e-6 / (((1.0 - z) - 1.0) + 7.0))) + (1.5056327351493116e-7 / (((1.0 - z) - 1.0) + 8.0))));
}
↓
double code(double z) {
	double t_0 = 676.5203681218851 / (1.0 - z);
	double t_1 = (1.0 - z) - 1.0;
	double t_2 = t_1 + 7.0;
	double t_3 = fma(t_0, (t_0 + -0.9999999999998099), 0.9999999999996197);
	return (((double) M_PI) / sin(((double) M_PI) * z)) * (((sqrt(((double) M_PI) * 2.0) * pow((t_2 + 0.5), (0.5 + log(exp(1.0 - z) / ((double) M_E))))) * exp(-0.5 - t_2)) * (((((((expm1(log1p(fma((2.0 - z), (0.9999999999994297 + pow(t_0, 3.0)), (-1259.1392167224028 * t_3)))) / ((2.0 - z) * t_3)) + (771.3234287776531 / (t_1 + 3.0))) + (-176.6150291621406 / (t_1 + 4.0))) + (12.507343278686905 / (t_1 + 5.0))) + (-0.13857109526572012 / (t_1 + 6.0))) + (9.984369578019572e-6 / t_2)) + (1.5056327351493116e-7 / (t_1 + 8.0))));
}

Error

Bits error versus z

Derivation

  1. Initial program 1.7

    \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

  2. Applied flip3-+_binary641.7

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\color{blue}{\frac{{0.9999999999998099}^{3} + {\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}}{0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}} + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

  3. Applied frac-add_binary641.0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\frac{\left({0.9999999999998099}^{3} + {\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 2\right) + \left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) \cdot -1259.1392167224028}{\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 2\right)}} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

  4. Simplified1.0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\color{blue}{\mathsf{fma}\left(2 - z, 0.9999999999994297 + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}, -1259.1392167224028 \cdot \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, 0.9999999999996197\right)\right)}}{\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 2\right)} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

  5. Simplified1.0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\mathsf{fma}\left(2 - z, 0.9999999999994297 + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}, -1259.1392167224028 \cdot \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, 0.9999999999996197\right)\right)}{\color{blue}{\left(2 - z\right) \cdot \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, 0.9999999999996197\right)}} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

  6. Applied expm1-log1p-u_binary640.7

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2 - z, 0.9999999999994297 + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}, -1259.1392167224028 \cdot \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, 0.9999999999996197\right)\right)\right)\right)}}{\left(2 - z\right) \cdot \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, 0.9999999999996197\right)} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

  7. Applied add-log-exp_binary640.7

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - \color{blue}{\log \left(e^{1}\right)}\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2 - z, 0.9999999999994297 + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}, -1259.1392167224028 \cdot \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, 0.9999999999996197\right)\right)\right)\right)}{\left(2 - z\right) \cdot \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, 0.9999999999996197\right)} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

  8. Applied add-log-exp_binary640.7

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\color{blue}{\log \left(e^{1 - z}\right)} - \log \left(e^{1}\right)\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2 - z, 0.9999999999994297 + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}, -1259.1392167224028 \cdot \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, 0.9999999999996197\right)\right)\right)\right)}{\left(2 - z\right) \cdot \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, 0.9999999999996197\right)} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

  9. Applied diff-log_binary640.7

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\color{blue}{\log \left(\frac{e^{1 - z}}{e^{1}}\right)} + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2 - z, 0.9999999999994297 + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}, -1259.1392167224028 \cdot \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, 0.9999999999996197\right)\right)\right)\right)}{\left(2 - z\right) \cdot \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, 0.9999999999996197\right)} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

  10. Final simplification0.7

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(0.5 + \log \left(\frac{e^{1 - z}}{e}\right)\right)}\right) \cdot e^{-0.5 - \left(\left(\left(1 - z\right) - 1\right) + 7\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2 - z, 0.9999999999994297 + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}, -1259.1392167224028 \cdot \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, 0.9999999999996197\right)\right)\right)\right)}{\left(2 - z\right) \cdot \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, 0.9999999999996197\right)} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

Reproduce

herbie shell --seed 2022067 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  :precision binary64
  :pre (<= z 0.5)
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5) (+ (- (- 1.0 z) 1.0) 0.5))) (exp (- (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1.0 z) 1.0) 1.0))) (/ -1259.1392167224028 (+ (- (- 1.0 z) 1.0) 2.0))) (/ 771.3234287776531 (+ (- (- 1.0 z) 1.0) 3.0))) (/ -176.6150291621406 (+ (- (- 1.0 z) 1.0) 4.0))) (/ 12.507343278686905 (+ (- (- 1.0 z) 1.0) 5.0))) (/ -0.13857109526572012 (+ (- (- 1.0 z) 1.0) 6.0))) (/ 9.984369578019572e-6 (+ (- (- 1.0 z) 1.0) 7.0))) (/ 1.5056327351493116e-7 (+ (- (- 1.0 z) 1.0) 8.0))))))